Solve for $x$ : $ 7|x + 5| + 9 = -5|x + 5| + 10 $
Explanation: Add $ {5|x + 5|} $ to both sides: $ \begin{eqnarray} 7|x + 5| + 9 &=& -5|x + 5| + 10 \\ \\ { + 5|x + 5|} && { + 5|x + 5|} \\ \\ 12|x + 5| + 9 &=& 10 \end{eqnarray} $ Subtract ${9}$ from both sides: $ \begin{eqnarray} 12|x + 5| + 9 &=& 10 \\ \\ { - 9} &=& { - 9} \\ \\ 12|x + 5| &=& 1 \end{eqnarray} $ Divide both sides by ${12}$ $ \dfrac{12|x + 5|} {{12}} = \dfrac{1} {{12}} $ Simplify: $ |x + 5| = \dfrac{1}{12}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 5 = -\dfrac{1}{12} $ or $ x + 5 = \dfrac{1}{12} $ Solve for the solution where $x + 5$ is negative: $ x + 5 = -\dfrac{1}{12} $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& -\dfrac{1}{12} \\ \\ {- 5} && {- 5} \\ \\ x &=& -\dfrac{1}{12} - 5 \end{eqnarray} $ Change the ${ - 5}$ to an equivalent fraction with a denominator of $12$ $ x = - \dfrac{1}{12} {- \dfrac{60}{12}} $ $ x = -\dfrac{61}{12} $ Then calculate the solution where $x + 5$ is positive: $ x + 5 = \dfrac{1}{12} $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& \dfrac{1}{12} \\ \\ {- 5} && {- 5} \\ \\ x &=& \dfrac{1}{12} - 5 \end{eqnarray} $ Change the ${ - 5}$ to an equivalent fraction with a denominator of $12$ $ x = \dfrac{1}{12} {- \dfrac{60}{12}} $ $ x = -\dfrac{59}{12} $ Thus, the correct answer is $x = -\dfrac{61}{12} $ or $x = -\dfrac{59}{12} $.